`2N_2O_5(g)rarr4NO_2(g)+O_2(g)` what is the ratio of the rate of decomposition of `N_2O_5` to rate of formation of `O_2` ?

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Answer

To find the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( O_2 \) in the reaction:

\[ 2N_2O_5(g) \rightarrow 4NO_2(g) + O_2(g) \]

we can follow these steps:

### Step 1: Write the Rate Expressions
For the given reaction, we can express the rates of change of concentrations as follows:

- The rate of decomposition of \( N_2O_5 \) is given by:
\[ \text{Rate of decomposition of } N_2O_5 = -\frac{1}{2} \frac{d[N_2O_5]}{dt} \]
(The negative sign indicates that the concentration of \( N_2O_5 \) is decreasing.)

- The rate of formation of \( O_2 \) is given by:
\[ \text{Rate of formation of } O_2 = \frac{1}{1} \frac{d[O_2]}{dt} \]

### Step 2: Relate the Rates
From the stoichiometry of the reaction, we can relate the rates of decomposition and formation. The coefficients in the balanced equation indicate how the rates are related:

\[ - \frac{1}{2} \frac{d[N_2O_5]}{dt} = \frac{1}{1} \frac{d[O_2]}{dt} \]

### Step 3: Set Up the Ratio
We need to find the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( O_2 \):

\[ \text{Ratio} = \frac{-\frac{1}{2} \frac{d[N_2O_5]}{dt}}{\frac{d[O_2]}{dt}} \]

### Step 4: Substitute the Rate Relationships
Using the relationship from Step 2, we can substitute:

\[ \text{Ratio} = \frac{-\frac{1}{2} \frac{d[N_2O_5]}{dt}}{\frac{1}{2} \frac{d[N_2O_5]}{dt}} = \frac{2}{1} \]

### Step 5: Final Result
Thus, the ratio of the rate of decomposition of \( N_2O_5 \) to the rate of formation of \( O_2 \) is:

\[ \text{Ratio} = 2:1 \]